recursive conditioning adnan darwiche computer science department ucla

Recursive Conditioning

Adnan DarwicheComputer Science Department

UCLA

Decomposition

Battery Age Alternator Fan Belt

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DecompositionBattery Age Alternator Fan Belt


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LP RP*

Causal Network



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Case AnalysisBattery Age Alternator Fan Belt

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Case I


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Case II


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Case I Case II

LP * RP


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Case I Case II

LP * RP LP * RP+



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LP * RP LP * RP+


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• Decomposition and Case Analysis can answer any query

• Non-Deterministic!

Decomposition Tree

A B C D E

A A B B C

C DD

BE

B

Decomposition Tree

A B C D E

A A B C

C DD

E

B

Decomposition Tree

A B C D E

A A B B C

C DD

BE

B

Decomposition Tree

A B C D E

A A B C

C DD

E

BLP RP

Decomposition Tree

A B C D E

A A B B C

C DD

BE

B LP * RP

Decomposition Tree

A B C D E

A A B C

C DD

E

B LP * RP

LP RP

Decomposition Tree

A B C D E

A A B B C

C DD

BE

B LP * RP LP * RP

Decomposition Tree

A B C D E

A A B B C

C DD

BE

LP * RP + LP * RP

Dtrees

Generation from– Elimination orders– Jointree– hMeTiS hypergraph

partitioning program

a a’

.60 .40

b b’

a .90 .10

a’ .20 .80

d d’

c .50 .50

c’ .40 .60

e e’

bd .25 .75

bd’ .25 .75

b’d .90 .10

b’d’ .05 .95

c c’

b .70 .30

b’ .15 .85

A B C D

E

A C

B

D

• Cutset

B B

BC

• Context• Cluster

• Vars

A AB

AB BCDE

BC

CD BDE

BCDE

RC1

RC1(T,e)if T is a leaf node

return Lookup(T,e)

else p := 0

for each instantiation c of cutset(T)-E dop := p + RC1(Tl,ec) RC1(Tr,ec)

return p

Lookup(T,e)X|U : CPT associated with leaf T

If X is instantiated in e, thenx: value of X in e

u: value of U in e

Return x|u

Else return 1 = xx|u

Computational Complexity

A B C D E

A A B B C

C DD

BE

A

B

C

D

B

BC

BCDAncestoral Cutset

Computational Complexity• Given

– DAG with n nodes– elimination order of width w

• Can construct a dtree in O(n log n) time:– Height O(log n)– cutset width <= w+1– a-cutset width O(w log n)

• Time complexity: O(n cw log n)

• Cutset Conditioning: O(n cs)

Network Effective Network

Size

Elimination-Order Width

Loop-Cutset Width

A-Cutset Width

1 Water 59.0 21.3 29.5 32.32 Midlew 108.4 19.7 39.3 45.93 Barley 138.0 21.8 57.3 51.14 Diabetes 1349.1 19.2 557.2 77.95 Link 922.7 28.0 347.0 70.76 Pigs 699.0 16.4 144.2 38.07 Munin1 410.3 26.4 122.6 51.78 Munin2 2202.7 20.0 495.9 54.69 Munin3 2293.4 16.3 454.2 51.1

10 Munin4 2313.4 19.7 521.5 51.9

Relation to Jointrees

Time: O(n cw

)

Space: O(n)

Time: O(n cwlog n

)

Space: O(n cw

)

Decomposition Tree

A B C D E F

A

A B

B C

C D

D E E F

AB

CA B C

Decomposition Tree

A B C D E F

A

A B

B C

C D

D E E F

AB

C CC

.27

.39

ABCABCABCABC

ABC

ABCABC

ABC

A B C

Context(N)= A-Cutset(N)&Vars(N)

Decomposition Tree

A B C D E F

A

A B

B C

C D

D E E F

AB

C

D

E

A

B

C

D

RC2

RC2(T,e)if T is a leaf node, return Lookup(T,e)

y := instantiation of context(T)

If cacheT[y] <> nil, return cacheT[y]

p := 0

for each instantiation c of cutset(T)-E dop := p + RC2(Tl,ec) RC2(Tr,ec)

cacheT[y] := p

return p

Constructing dtreesA

D

B

E

C

F

F E A B C D

A AB AC ABE BCD DF

DBC

A

W=2

Cutset <= w+1


D

B

E

C

F

F E A B C D

A AB AC ABE BCD DF

DBC

A

W=2


D

B

E

C

F

F E A B C D

A AB AC ABE BCD DF

DBC

A

W=2

DBC

AB ABC

Context <= w+1

Space: O(n cw

)


D

B

E

C

F

F E A B C D

A AB AC ABE BCD DF

D

BCDABC

W=2

AB ABC

Cluster<= w+1

Space: O(n cw

)Time: O(n c

w)


D

B

E

C

F

F E A B C D

ABE DF

BCDABC

W=2

Cluster<= w+1

Graphical Models

Elimination Order Jointree

Dtree

Hypergraph partitioningHypergraph partitioning

The problem of hypergraphpartitioning is well-studied

in VLSI design.….and is alive!

A hypergraph is a generalization of a graph, such that an edge is permitted to connect an arbitrary number of vertices, rather than exactly two.

The task of hypergraph partitioning is to find a way to split the vertices of a hypergraph into k approximately equal parts, such that the number of hyperedges connecting vertices in different parts is minimized.

An algorithm using hypergraph An algorithm using hypergraph partitioning to construct dtreespartitioning to construct dtrees

Given a DAG G, we first generate a hypergraph H that corresponds to G:

For each family F in G, we add a node NF to H. For each variable V in G, we add a hyperedge to H which connects all nodes NF such that V is a member of F.

Then we recursively invoke hypergraph partitioning to build a dtree:

The algorithm partitions the hypergraph into two sets of vertices, then recursively generates dtrees for each set, and finally combines the resulting dtrees into a new dtree (whose left child is the dtree for the first set, and whose right child is the dtree for the second set).

A AB

CDBC

A AB CDBC

A AB CDBC

For DAG:

A

CB

ED

F

HG

DFDF

AC ACE

EFH A AF

AB ABD

DFG

CC

EEHH

AAGG

BB

From dtrees to elimination ordersFrom dtrees to elimination orders

Any-Space Inference

32

8 16

8

64 128

1024 512

1728 cache entries 64 cache entries

Time-Space TradeoffsTime-Space Tradeoffs

.8

0 1

.45

Cache FactorsCache Factors

.9 1 0

0

cf(T) : [0,1]

RC

RC(T,e)if T is a leaf node, return Lookup(T,e)

y := instantiation of context(T)

If cacheT[y] <> nil, return cacheT[y]

p := 0

for each instantiation c of cutset(T)-E dop := p + RC(Tl,ec) RC(Tr,ec)

If cacheT[y]? then cacheT[y] := p

return p

calls(T)=cutset#(Tp) [cf(Tp) context#(Tp) + (1-cf(Tp)) calls(Tp)]

When cf(T)=0 for every node T:

calls(T) = cutset#(Tp) calls(Tp)

When cf(T)=1 for every node T:

calls(T) = cutset#(Tp) context#(Tp) = cluster#(Tp)

exact count for discrete cache factors

Extension to Dgraphs

)(A )|( BD)|( BC)|( AB

Marginals

B

Dgraph

)(A )|( BD)|( BC)|( AB

AB

DgraphAB A

)(A )|( BD)|( BC)|( AB

Dgraph

)(A )|( BD)|( BC)|( AB

AB A BC

Dgraph

)(A )|( BD)|( BC)|( AB

AB A BC BD

Number of Dtree Internal Nodes

Network Dtree Dgraph

Water 31 122

Alarm 36 142

B 17 66

Barley 47 186

SamIam Screenshot

Discrete Cache Factor Search

• Problem– Given X MB of cache memory for RC– Find a discrete caching scheme which has a

minimal running time where the amount of memory used by all the dtree node caches is no larger than X

Number of RC Calls

• Can estimate running time of RC

• Function of: – dtree– cache factors

pt

ppppp tcallstcftcontexttcftcutsettcalls *1**)(##

Different Cache Factors

1

2 3

4 5 6 7

cf=0

cf=1cf=01

2 3

4 5 6 7

cf=0

cf=0cf=0

1

2 3

4 5 6 7

cf=1

cf=0cf=01

2 3

4 5 6 7

cf=0

cf=1cf=1

1

2 3

4 5 6 7

cf=0

cf=0cf=1

1

2 3

4 5 6 7

cf=1

cf=1cf=0

1

2 3

4 5 6 7

cf=1

cf=0cf=11

2 3

4 5 6 7

cf=1

cf=1cf=1

20 sec65 sec35 sec

75 sec80 sec90 sec

30 sec 10 sec

Search Tree

1

2 3

4 5 6 7

cf=?

cf=?cf=?

1

2 3

4 5 6 7

cf=0

cf=?cf=?

No Caching

……

1

2 3

4 5 6 7

cf=1

cf=?cf=?

Full Caching

……

Cost Function

1

2 3

4 5 6 7

cf=?

cf=?cf=?

1

2 3

4 5 6 7

cf=0

cf=?cf=?1

2 3

4 5 6 7

cf=1

cf=?cf=?

… …… …

Cost Function

1

2 3

4 5 6 7

cf=?

cf=?cf=?

1

2 3

4 5 6 7

cf=0

cf=?cf=?1

2 3

4 5 6 7

cf=1

cf=1cf=1

… …… …

Admissible

Search Tree Pruning

• Memory-based pruning– Prune any branch which assigns more than the

total memory assigned to the algorithm

• Cost-based pruning– Prune any node whose cost function f(n) is

already larger than the current best goal node

3.3 MB = 28 Sec7.6 MB = 12 Sec

.56 MB = 59 Sec

2.6 MB = 5.9 Sec

1 MB = 15.7 Sec

.06 MB = 120 Sec

.19 KB = 2 Sec2.8 KB < .01 Sec.23 KB = .7 Sec

22 MB = 1 Min

6.5 MB = 4 Min

370 MB = 22 Min

150 MB = 5.8 Hours

400 MB = 18 Min

75 MB = 2 Hours

Conclusion: Inference

• Alternative conditioning paradigm– condition to decompose– dtrees: decomposition policy

• New complexity results:– linear space: O(n cwlog n)

– any-space: O(n) ---- O(n cw)

O(n cwlog n) ---- O(n cw)

recursive conditioning adnan darwiche computer science department ucla

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